Problem 13 - Algebra A Combined Function

Question

Add or subtract as indicated. Assume that all variables represent positive real numbers. $$ \sqrt[3]{\frac{11}{8}}-\frac{\sqrt[3]{11}}{6} $$

Step-by-Step Solution

Answer

\( \frac{\sqrt[3]{11}}{3} \)

Step 1: Identify the expression

The expression given is \( \sqrt[3]{\frac{11}{8}} - \frac{\sqrt[3]{11}}{6} \). We need to find a common form to combine these terms if possible.

Step 2: Simplify each component

First, recall that \( \sqrt[3]{\frac{11}{8}} \) can be expressed as \( \frac{\sqrt[3]{11}}{\sqrt[3]{8}} \). And since \( \sqrt[3]{8} = 2 \), we have \( \frac{\sqrt[3]{11}}{2} \). The expression now reads \( \frac{\sqrt[3]{11}}{2} - \frac{\sqrt[3]{11}}{6} \).

Step 3: Find a common denominator

The fractions \( \frac{\sqrt[3]{11}}{2} \) and \( \frac{\sqrt[3]{11}}{6} \) need a common denominator to be combined. The least common denominator of 2 and 6 is 6. Therefore, we'll rewrite \( \frac{\sqrt[3]{11}}{2} \) as \( \frac{3\sqrt[3]{11}}{6} \).

Step 4: Subtract the fractions

Now that both fractions have the same denominator, combine them: \( \frac{3\sqrt[3]{11}}{6} - \frac{\sqrt[3]{11}}{6} = \frac{(3\sqrt[3]{11} - \sqrt[3]{11})}{6} \).

Step 5: Simplify the numerator

Factor out \( \sqrt[3]{11} \) from the numerator: \( 3\sqrt[3]{11} - \sqrt[3]{11} = (3 - 1)\sqrt[3]{11} = 2\sqrt[3]{11} \).

Step 6: Final Answer

Substitute the simplified numerator back into the fraction: \( \frac{2\sqrt[3]{11}}{6} \). Simplify the fraction by dividing both the numerator and the denominator by 2 to get \( \frac{\sqrt[3]{11}}{3} \).

Key Concepts

Cube Root Simplification

Simplifying cube roots is an essential skill when dealing with expressions involving square roots. The cube root of a number is a value that, when multiplied by itself three times, gives the original number back. For instance, the cube root of 8 is 2 because

• 2 × 2 × 2 equals 8.

To simplify the cube root of a fraction, like \( \sqrt[3]{\frac{11}{8}} \), we can break it down into two separate roots:

• \( \sqrt[3]{11} \) divided by \( \sqrt[3]{8} \).

Since we know that \( \sqrt[3]{8} = 2 \), this expression becomes \( \frac{\sqrt[3]{11}}{2} \). This is a more manageable form that can be useful for further operations like addition, subtraction, or multiplication, especially when combining multiple terms.

Finding Common Denominator

When you're dealing with fractions, finding a common denominator is critical for combining or comparing them. Here, we have two fractions \( \frac{\sqrt[3]{11}}{2} \) and \( \frac{\sqrt[3]{11}}{6} \) that need to be subtracted from each other. They're not directly comparable because they have different denominators.

• The denominators are 2 and 6.

A common method is to find the least common denominator (LCD). For 2 and 6, the smallest number that both these denominators divide into is 6. Therefore, 6 is the LCD.To convert \( \frac{\sqrt[3]{11}}{2} \) into an equivalent fraction with a denominator of 6, multiply both the numerator and denominator by 3, resulting in \( \frac{3\sqrt[3]{11}}{6} \). This step ensures both fractions can now be subtracted directly because they share the same denominator. This concept is paramount when performing arithmetic operations on fractions.

Fraction Subtraction

Once the fractions \( \frac{3\sqrt[3]{11}}{6} \) and \( \frac{\sqrt[3]{11}}{6} \) have a common denominator, subtracting them becomes straightforward. Think of fraction subtraction much like subtracting simple numbers; keep the denominator the same and subtract the numerators.Here's how to do it:

• Numerator subtraction: \( 3\sqrt[3]{11} - \sqrt[3]{11} \).
• This simplifies to \( (3 - 1)\sqrt[3]{11} = 2\sqrt[3]{11} \).

The resulting expression is \( \frac{2\sqrt[3]{11}}{6} \). Finish by simplifying it. Since both the numerator and the denominator are divisible by 2, the fraction simplifies to \( \frac{\sqrt[3]{11}}{3} \). Simplification in this step is crucial because it gives us the answer in its most reduced form, which is often required in mathematical problems.